Problem: Let $r$ be a complex number such that $r^5 = 1$ and $r \neq 1.$  Compute
\[(r - 1)(r^2 - 1)(r^3 - 1)(r^4 - 1).\]
We can write $r^5 - 1 = 0,$ which factors as
\[(r - 1)(r^4 + r^3 + r^2 + r + 1) = 0.\]Since $r \neq 1,$ $r^4 + r^3 + r^2 + r + 1 = 0.$

To compute the product, we can arrange the factors in pairs:
\begin{align*}
(r - 1)(r^2 - 1)(r^3 - 1)(r^4 - 1) &= [(r - 1)(r^4 - 1)][(r^2 - 1)(r^3 - 1)] \\
&= (r^5 - r - r^4 + 1)(r^5 - r^2 - r^3 + 1) \\
&= (1 - r - r^4 + 1)(1 - r^2 - r^3 + 1) \\
&= (2 - r - r^4)(2 - r^2 - r^3) \\
&= 4 - 2r^2 - 2r^3 - 2r + r^3 + r^4 - 2r^4 + r^6 + r^7 \\
&= 4 - 2r^2 - 2r^3 - 2r + r^3 + r^4 - 2r^4 + r + r^2 \\
&= 4 - r - r^2 - r^3 - r^4 \\
&= 5 - (1 + r + r^2 + r^3 + r^4) = \boxed{5}.
\end{align*}